What kind of AI technique can I use to play the "Lines" game?
Writing a game AI is difficult in general and it's even harder when there's no game to play. Like in your case.

While "Lines" is pretty neat, there's no (playing) opponent and there are no moves. apart from the setup. So what we have is an optimization problem where we need to get more point than the "opponents".

Game playing algorithms like Reinforcement learning or MCTS are IMHO out of questions as there's no game to play. Still, it's an interesting problem, though I can't see how to make any learning AI. Anyway, what you need first is a simulator and an optimizer choosing promising actions.

Some ideas concerning the actions

Some actions are trivial, e.g., the eraser as there are just a few dots to erase. Trying all combination should be good enough for the start.

The scissors can be in theory placed anywhere, but the only interesting places are just next to the oponent piece or a crossing. Additionally, you only need to consider crossings which gets "won" (reached first) by an opponent. Moreover, they only matter if they get won by an opponent having more points than you or blocking you.

The dots to insert should similarly by placed near an oponent or so that you just win a crossing.

The additional lines are worst, as they may be used to let you reach unconnected areas, or reach a crossing faster or simply give you additional points (then you want a long line somewhere).

A model

Most of the game can be seen as a non-directed graph with edges labeled by their length, where the crossing and the initial dots are nodes. The simulator doesn't need to work "pixel by pixel", but may instead jump to the next time, when a crossing gets reached or two "flows" meat each other. The points of a player grow linearly with time, where the slope is given by the number of their extending "flows".

So far, I could imagine modelling it as an Integer Linear Program, but with the additional lines, it's probably no more possible.

I'd probably let the optimization be simulator-guided, i.e., analyse whether I could get closer to winning by reaching a crossing sooner. The details are still unclear....